Abstract
This article formulates two variants of packing problems in which the set of items is known, but the arrival order is unknown. The goal is to certify that the items can be packed in a given container and/or to optimize the size or cost of a container so that that the items are guaranteed to be packable, regardless of arrival order. The nondeterministically ordered packing (NDOP) variant asks to generate a certificate that a packing plan exists for every ordering of items. Quasi-online packing (QOP) asks to generate a partially observable packing policy that chooses the location of each item as their arrival order is revealed one-by-one, with all of the remaining items certified to be packable regardless of their future arrival order. Theoretical analysis demonstrates that even the simple subproblem of verifying the feasibility of a packing policy is NP-complete. Despite this worst case complexity, practical solvers for both NDOP and QOP are developed. Multiple extensions to the basic nondeterministic problem are presented, including packing with a fixed-capacity buffer and packing with equivalent objects. Experiments demonstrate that these algorithms can be applied to packing irregular 3-D shapes with stability and manipulator loading constraints. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Automatic packing of objects into containers, such as a shipping box, has applications in warehouse automation for e-commerce applications and less-than-truckload shipping. In many scenarios, a container needs to be preselected for a set of objects before they arrive, and the order of the objects cannot be controlled. This article studies the theoretical foundations of packing problems, in which the set of items is known, but the arrival order is unknown. The algorithms presented in this article can certify whether a container can hold a set of objects in the case where the arrival order is revealed at once, one-by-one, and where a limited set of objects can be put aside in a buffer before packing. These algorithms can be used to choose optimal containers and packing strategies, both for robot and human packers.
Highlights
I NTEREST in warehouse automation has grown rapidly with the growth of e-commerce and advances in robotics
We prove that the worst case solution complexity of nondeterministically ordered packing (NDOP) and Quasionline packing (QOP) is O(n!), and even feasibility verification for a polynomial-sized NDOP policy is NP-complete via a reduction from Boolean satisfiability (SAT)
A key subroutine used in our NDOP and QOP planners verifies whether a set of packing plans P1, . . . , Pm is compatible with all orderings in Sn, and if not, to generate a
Summary
I NTEREST in warehouse automation has grown rapidly with the growth of e-commerce and advances in robotics. The items and container(s) are known, and a plan can place the items in arbitrary order [2]. This article introduces two nondeterministic formulations of robot packing problems that lie between the offline and online settings. These formulations are practical for automated warehouses where the ultimate item set (e.g., shopping cart) is known, but some distinct, uncontrollable component of the packing system controls the item arrival order. In Amazon’s automated fulfillment centers, shelving units containing individual items are carried by thousands of mobile robots to several picking stations, and the order in which shelves arrive at a given station is controlled by a complex algorithm that is tuned to maximize delivery throughput for shelving units. Larger buffers allow for denser packing in the worst case, which has consequences for workcell design
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